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        <p id="owner"><a name="bijectiveGoedelNumberings" class="extype" href="package.html">bijectiveGoedelNumberings</a></p>
        <h1>FTermTester</h1>
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        <span class="name">FTermTester</span><span class="result"> extends <a name="bijectiveGoedelNumberings.FGoedelNums" class="extype" href="FGoedelNums.html">FGoedelNums</a>[String, String]</span>
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      <div class="fullcommenttop" id="comment"><div class="comment cmt"><p>Various tests for Goedel numberings of fixed signature terms.
Note that we instantiate the type parameter A and B
to the specific type of function, constant and variable
symbols and we define specific finite Arrays of each.
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      <a id="binomial(BigInt,BigInt):BigInt"></a>
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        <span class="name">binomial</span><span class="params">(<span name="n">n: BigInt</span>, <span name="r0">r0: BigInt</span>)</span><span class="result">: BigInt</span>
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      <p class="shortcomment cmt">Computes the binomial &quot;n choose k&quot;
</p><div class="fullcomment"><div class="comment cmt"><p>Computes the binomial &quot;n choose k&quot;
</p></div><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="bijectiveGoedelNumberings.Catalan" class="extype" href="Catalan.html">Catalan</a></dd></dl></div>
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        <span class="modifier"></span>
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        <span class="name">fbigt</span><span class="params">(<span name="n">n: BigInt</span>)</span><span class="result">: <a name="bijectiveGoedelNumberings.FixedSignatureTerm" class="extype" href="FixedSignatureTerm.html">FixedSignatureTerm</a>[String, String]</span>
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      <p class="shortcomment cmt">Generates a term of representation size O(n).</p><div class="fullcomment"><div class="comment cmt"><p>Generates a term of representation size O(n).
Tests that unique natural numbers correspond to any terms.
</p></div></div>
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      </h4>
      <p class="shortcomment cmt">Tests behavior on big terms
</p>
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        <span class="name">firstKBinomialLargerThan</span><span class="params">(<span name="k">k: BigInt</span>, <span name="n">n: BigInt</span>)</span><span class="result">: BigInt</span>
      </span>
      </h4>
      <p class="shortcomment cmt"> efficently finds the smallest m such that (m choose k &gt; n) using
 binary search
</p><div class="fullcomment"><div class="comment cmt"><p> efficently finds the smallest m such that (m choose k &gt; n) using
 binary search
</p></div><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="bijectiveGoedelNumberings.Cantor" class="extype" href="Cantor.html">Cantor</a></dd></dl></div>
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        <span class="name">fromCantorTuple</span><span class="params">(<span name="ns">ns: List[BigInt]</span>)</span><span class="result">: BigInt</span>
      </span>
      </h4>
      <p class="shortcomment cmt">Computes the natural number uniquely associated
to the sequence of natural numbers ns, using
Cantor's generalized bijection formula
</p><div class="fullcomment"><div class="comment cmt"><p>Computes the natural number uniquely associated
to the sequence of natural numbers ns, using
Cantor's generalized bijection formula
</p></div><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="bijectiveGoedelNumberings.Cantor" class="extype" href="Cantor.html">Cantor</a></dd></dl></div>
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        <span class="name">ftest</span><span class="params">(<span name="m">m: <span name="scala.Int" class="extype">Int</span></span>, <span name="k">k: <span name="scala.Int" class="extype">Int</span></span>)</span><span class="result">: <span name="scala.Unit" class="extype">Unit</span></span>
      </span>
      </h4>
      <p class="shortcomment cmt">Generates terms associated to a range of natural numbers.</p><div class="fullcomment"><div class="comment cmt"><p>Generates terms associated to a range of natural numbers.
Tests the fact that all natural numbers correspond
to terms.
</p></div></div>
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    </li><li visbl="pub" name="bijectiveGoedelNumberings.FGoedelNums#get_bdigit" data-isabs="false">
      <a id="get_bdigit(BigInt,BigInt):(BigInt, BigInt)"></a>
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      </span>
      </h4>
      <p class="shortcomment cmt">extract one bijective base-b digit from
natural number n
</p><div class="fullcomment"><div class="comment cmt"><p>extract one bijective base-b digit from
natural number n
</p></div><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="bijectiveGoedelNumberings.FGoedelNums" class="extype" href="FGoedelNums.html">FGoedelNums</a></dd></dl></div>
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    </li><li visbl="pub" name="bijectiveGoedelNumberings.FGoedelNums#nat2term" data-isabs="false">
      <a id="nat2term(BigInt):FixedSignatureTerm[String, String]"></a>
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      <span class="modifier_kind">
        <span class="modifier"></span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">nat2term</span><span class="params">(<span name="n">n: BigInt</span>)</span><span class="result">: <a name="bijectiveGoedelNumberings.FixedSignatureTerm" class="extype" href="FixedSignatureTerm.html">FixedSignatureTerm</a>[String, String]</span>
      </span>
      </h4>
      <p class="shortcomment cmt">Builds the unique term associated to
natural number n, using symbols from the arrays
&quot;consts&quot; and &quot;funs&quot; and variables from the
array &quot;vars&quot;.</p><div class="fullcomment"><div class="comment cmt"><p>Builds the unique term associated to
natural number n, using symbols from the arrays
&quot;consts&quot; and &quot;funs&quot; and variables from the
array &quot;vars&quot;.</p><p>The algorithm works by associating
small numbers to variable and constant symbols
and then it uses the generalized Cantor bijection
toCantorTuple to split larger numbers into
sequences matching the arities of the
function symbols, from which it creates
subterms recursively.
</p></div><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="bijectiveGoedelNumberings.FGoedelNums" class="extype" href="FGoedelNums.html">FGoedelNums</a></dd></dl></div>
    </li><li visbl="pub" name="scala.AnyRef#ne" data-isabs="false">
      <a id="ne(AnyRef):Boolean"></a>
      <h4 class="signature">
      <span class="modifier_kind">
        <span class="modifier">final </span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">ne</span><span class="params">(<span name="arg0">arg0: AnyRef</span>)</span><span class="result">: <span name="scala.Boolean" class="extype">Boolean</span></span>
      </span>
      </h4>
      <div class="fullcomment"><dl class="attributes block"> <dt>Definition Classes</dt><dd>AnyRef</dd></dl></div>
    </li><li visbl="pub" name="scala.AnyRef#notify" data-isabs="false">
      <a id="notify():Unit"></a>
      <h4 class="signature">
      <span class="modifier_kind">
        <span class="modifier">final </span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">notify</span><span class="params">()</span><span class="result">: <span name="scala.Unit" class="extype">Unit</span></span>
      </span>
      </h4>
      <div class="fullcomment"><dl class="attributes block"> <dt>Definition Classes</dt><dd>AnyRef</dd></dl></div>
    </li><li visbl="pub" name="scala.AnyRef#notifyAll" data-isabs="false">
      <a id="notifyAll():Unit"></a>
      <h4 class="signature">
      <span class="modifier_kind">
        <span class="modifier">final </span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">notifyAll</span><span class="params">()</span><span class="result">: <span name="scala.Unit" class="extype">Unit</span></span>
      </span>
      </h4>
      <div class="fullcomment"><dl class="attributes block"> <dt>Definition Classes</dt><dd>AnyRef</dd></dl></div>
    </li><li visbl="pub" name="bijectiveGoedelNumberings.FGoedelNums#put_bdigit" data-isabs="false">
      <a id="put_bdigit(BigInt,BigInt,BigInt):BigInt"></a>
      <h4 class="signature">
      <span class="modifier_kind">
        <span class="modifier"></span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">put_bdigit</span><span class="params">(<span name="b">b: BigInt</span>, <span name="d">d: BigInt</span>, <span name="m">m: BigInt</span>)</span><span class="result">: <span name="scala.math.BigInt" class="extype">BigInt</span></span>
      </span>
      </h4>
      <p class="shortcomment cmt">integrates bijective base-b digit d into natural
number m
</p><div class="fullcomment"><div class="comment cmt"><p>integrates bijective base-b digit d into natural
number m
</p></div><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="bijectiveGoedelNumberings.FGoedelNums" class="extype" href="FGoedelNums.html">FGoedelNums</a></dd></dl></div>
    </li><li visbl="pub" name="bijectiveGoedelNumberings.Catalan#rank" data-isabs="false">
      <a id="rank(List[Int]):BigInt"></a>
      <h4 class="signature">
      <span class="modifier_kind">
        <span class="modifier"></span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">rank</span><span class="params">(<span name="xs">xs: List[<span name="scala.Int" class="extype">Int</span>]</span>)</span><span class="result">: BigInt</span>
      </span>
      </h4>
      <p class="shortcomment cmt">Computes natural number uniquely associated
to given balanced parentehsis string xs
</p><div class="fullcomment"><div class="comment cmt"><p>Computes natural number uniquely associated
to given balanced parentehsis string xs
</p></div><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="bijectiveGoedelNumberings.Catalan" class="extype" href="Catalan.html">Catalan</a></dd></dl></div>
    </li><li visbl="pub" name="bijectiveGoedelNumberings.Cantor#set2list" data-isabs="false">
      <a id="set2list(List[BigInt]):List[BigInt]"></a>
      <h4 class="signature">
      <span class="modifier_kind">
        <span class="modifier"></span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">set2list</span><span class="params">(<span name="xs">xs: List[BigInt]</span>)</span><span class="result">: List[BigInt]</span>
      </span>
      </h4>
      <p class="shortcomment cmt">Bijection from finite sets of natural numbers
(canonically represented as non-decreasing sequences)
to finite sequences of natural numbers
</p><div class="fullcomment"><div class="comment cmt"><p>Bijection from finite sets of natural numbers
(canonically represented as non-decreasing sequences)
to finite sequences of natural numbers
</p></div><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="bijectiveGoedelNumberings.Cantor" class="extype" href="Cantor.html">Cantor</a></dd></dl></div>
    </li><li visbl="pub" name="bijectiveGoedelNumberings.Catalan#showpars" data-isabs="false">
      <a id="showpars(List[Int]):String"></a>
      <h4 class="signature">
      <span class="modifier_kind">
        <span class="modifier"></span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">showpars</span><span class="params">(<span name="xs">xs: List[<span name="scala.Int" class="extype">Int</span>]</span>)</span><span class="result">: <span name="java.lang.String" class="extype">String</span></span>
      </span>
      </h4>
      <div class="fullcomment"><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="bijectiveGoedelNumberings.Catalan" class="extype" href="Catalan.html">Catalan</a></dd></dl></div>
    </li><li visbl="pub" name="scala.AnyRef#synchronized" data-isabs="false">
      <a id="synchronized[T0](⇒ T0):T0"></a>
      <h4 class="signature">
      <span class="modifier_kind">
        <span class="modifier">final </span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">synchronized</span><span class="tparams">[<span name="T0">T0</span>]</span><span class="params">(<span name="arg0">arg0: ⇒ T0</span>)</span><span class="result">: T0</span>
      </span>
      </h4>
      <div class="fullcomment"><dl class="attributes block"> <dt>Definition Classes</dt><dd>AnyRef</dd></dl></div>
    </li><li visbl="pub" name="bijectiveGoedelNumberings.FGoedelNums#term2nat" data-isabs="false">
      <a id="term2nat(FixedSignatureTerm[String, String]):BigInt"></a>
      <h4 class="signature">
      <span class="modifier_kind">
        <span class="modifier"></span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">term2nat</span><span class="params">(<span name="t">t: <a name="bijectiveGoedelNumberings.FixedSignatureTerm" class="extype" href="FixedSignatureTerm.html">FixedSignatureTerm</a>[String, String]</span>)</span><span class="result">: BigInt</span>
      </span>
      </h4>
      <p class="shortcomment cmt">Returns the unique natural number (represented as a BigInt)
associated to a term t.</p><div class="fullcomment"><div class="comment cmt"><p>Returns the unique natural number (represented as a BigInt)
associated to a term t.
</p></div><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="bijectiveGoedelNumberings.FGoedelNums" class="extype" href="FGoedelNums.html">FGoedelNums</a></dd></dl></div>
    </li><li visbl="pub" name="bijectiveGoedelNumberings.Cantor#toCantorTuple" data-isabs="false">
      <a id="toCantorTuple(BigInt,BigInt):List[BigInt]"></a>
      <h4 class="signature">
      <span class="modifier_kind">
        <span class="modifier"></span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">toCantorTuple</span><span class="params">(<span name="k">k: BigInt</span>, <span name="n">n: BigInt</span>)</span><span class="result">: List[BigInt]</span>
      </span>
      </h4>
      <p class="shortcomment cmt">Computes a sequence of k integers uniquely associated
to n using Cantor's generalized bijection.</p><div class="fullcomment"><div class="comment cmt"><p>Computes a sequence of k integers uniquely associated
to n using Cantor's generalized bijection.</p><p>This is an effciently computed inverse of
fromCantorTuple - the key contribution
of Paul Tarau's iclp12 paper
</p></div><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="bijectiveGoedelNumberings.Cantor" class="extype" href="Cantor.html">Cantor</a></dd></dl></div>
    </li><li visbl="pub" name="bijectiveGoedelNumberings.Cantor#toCombinadics" data-isabs="false">
      <a id="toCombinadics(BigInt,BigInt):List[BigInt]"></a>
      <h4 class="signature">
      <span class="modifier_kind">
        <span class="modifier"></span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">toCombinadics</span><span class="params">(<span name="k">k: BigInt</span>, <span name="n">n: BigInt</span>)</span><span class="result">: List[BigInt]</span>
      </span>
      </h4>
      <p class="shortcomment cmt">Converts n to its &quot;combinadic&quot; representation
(see Knuth's AOCP fascicle for definitions)
</p><div class="fullcomment"><div class="comment cmt"><p>Converts n to its &quot;combinadic&quot; representation
(see Knuth's AOCP fascicle for definitions)
</p></div><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="bijectiveGoedelNumberings.Cantor" class="extype" href="Cantor.html">Cantor</a></dd></dl></div>
    </li><li visbl="pub" name="scala.AnyRef#toString" data-isabs="false">
      <a id="toString():String"></a>
      <h4 class="signature">
      <span class="modifier_kind">
        <span class="modifier"></span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">toString</span><span class="params">()</span><span class="result">: <span name="java.lang.String" class="extype">String</span></span>
      </span>
      </h4>
      <div class="fullcomment"><dl class="attributes block"> <dt>Definition Classes</dt><dd>AnyRef → Any</dd></dl></div>
    </li><li visbl="pub" name="bijectiveGoedelNumberings.Catalan#unrank" data-isabs="false">
      <a id="unrank(BigInt):List[Int]"></a>
      <h4 class="signature">
      <span class="modifier_kind">
        <span class="modifier"></span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">unrank</span><span class="params">(<span name="r">r: BigInt</span>)</span><span class="result">: List[<span name="scala.Int" class="extype">Int</span>]</span>
      </span>
      </h4>
      <p class="shortcomment cmt">Computes balanced parenthesis expression uniquely
associated to natural number r
</p><div class="fullcomment"><div class="comment cmt"><p>Computes balanced parenthesis expression uniquely
associated to natural number r
</p></div><dl class="attributes block"> <dt>Definition Classes</dt><dd><a name="bijectiveGoedelNumberings.Catalan" class="extype" href="Catalan.html">Catalan</a></dd></dl></div>
    </li><li visbl="pub" name="scala.AnyRef#wait" data-isabs="false">
      <a id="wait():Unit"></a>
      <h4 class="signature">
      <span class="modifier_kind">
        <span class="modifier">final </span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">wait</span><span class="params">()</span><span class="result">: <span name="scala.Unit" class="extype">Unit</span></span>
      </span>
      </h4>
      <div class="fullcomment"><dl class="attributes block"> <dt>Definition Classes</dt><dd>AnyRef</dd><dt>Annotations</dt><dd>
                <span class="name">@throws</span><span class="args">()</span>
              
        </dd></dl></div>
    </li><li visbl="pub" name="scala.AnyRef#wait" data-isabs="false">
      <a id="wait(Long,Int):Unit"></a>
      <h4 class="signature">
      <span class="modifier_kind">
        <span class="modifier">final </span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">wait</span><span class="params">(<span name="arg0">arg0: <span name="scala.Long" class="extype">Long</span></span>, <span name="arg1">arg1: <span name="scala.Int" class="extype">Int</span></span>)</span><span class="result">: <span name="scala.Unit" class="extype">Unit</span></span>
      </span>
      </h4>
      <div class="fullcomment"><dl class="attributes block"> <dt>Definition Classes</dt><dd>AnyRef</dd><dt>Annotations</dt><dd>
                <span class="name">@throws</span><span class="args">()</span>
              
        </dd></dl></div>
    </li><li visbl="pub" name="scala.AnyRef#wait" data-isabs="false">
      <a id="wait(Long):Unit"></a>
      <h4 class="signature">
      <span class="modifier_kind">
        <span class="modifier">final </span>
        <span class="kind">def</span>
      </span>
      <span class="symbol">
        <span class="name">wait</span><span class="params">(<span name="arg0">arg0: <span name="scala.Long" class="extype">Long</span></span>)</span><span class="result">: <span name="scala.Unit" class="extype">Unit</span></span>
      </span>
      </h4>
      <div class="fullcomment"><dl class="attributes block"> <dt>Definition Classes</dt><dd>AnyRef</dd><dt>Annotations</dt><dd>
                <span class="name">@throws</span><span class="args">()</span>
              
        </dd></dl></div>
    </li></ol>
            </div>

        
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        <div id="inheritedMembers">
        <div name="bijectiveGoedelNumberings.FGoedelNums" class="parent">
              <h3>Inherited from <a name="bijectiveGoedelNumberings.FGoedelNums" class="extype" href="FGoedelNums.html">FGoedelNums</a>[String, String]</h3>
            </div><div name="bijectiveGoedelNumberings.Cantor" class="parent">
              <h3>Inherited from <a name="bijectiveGoedelNumberings.Cantor" class="extype" href="Cantor.html">Cantor</a></h3>
            </div><div name="bijectiveGoedelNumberings.Catalan" class="parent">
              <h3>Inherited from <a name="bijectiveGoedelNumberings.Catalan" class="extype" href="Catalan.html">Catalan</a></h3>
            </div><div name="scala.AnyRef" class="parent">
              <h3>Inherited from AnyRef</h3>
            </div><div name="scala.Any" class="parent">
              <h3>Inherited from <span name="scala.Any" class="extype">Any</span></h3>
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